In this paper, we have used branch continuation to track two-dimensional, finite-amplitude travelling waves in a viscoelastic channel flow, using the FENE-P model.
The travelling waves are borne out of the centre mode instability [1]} which is known to be subcritical over large areas of the \(Re-Wi\)  parameter space [2]}.
Here, we have shown that the TW solution branches extend to significantly lower \(Re\)  and \(Wi\)  than the curve of marginal stability.
For instance, we showed that the saddle node at \(Re=1000\)  drops as low as \(Wi\approx 25\) , while the associated linear instability at this \(Re\)  does not occur until \(Wi\approx 80\) .
Most significantly, we demonstrated the persistence of the nonlinear TWs at \(Re=0\)  for a large range of \(Wi\) , with the saddle node sitting at \(Wi \approx 7.5-10\)  in dilute (\(\beta =0.9\) ) solutions for a range of \(L_{max}\) .
Across a broad range of the parameter space, including at \(Re=0\) , the upper branch TWs resemble the arrowhead structures observed in DNS at higher \(Re\)  [3]}, [4]}.
